Properties

Label 2-8470-1.1-c1-0-117
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s − 3·9-s + 10-s + 4·13-s + 14-s + 16-s − 4·17-s + 3·18-s − 4·19-s − 20-s − 2·23-s + 25-s − 4·26-s − 28-s + 4·29-s + 2·31-s − 32-s + 4·34-s + 35-s − 3·36-s + 6·37-s + 4·38-s + 40-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 0.917·19-s − 0.223·20-s − 0.417·23-s + 1/5·25-s − 0.784·26-s − 0.188·28-s + 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.169·35-s − 1/2·36-s + 0.986·37-s + 0.648·38-s + 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8470\)    =    \(2 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(67.6332\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.64934973439899225727472268030, −6.62894565150512462651818119117, −6.28798714251552491550955338860, −5.62171747222069653671744841533, −4.47651206452587388206130850645, −3.85325849305816087434107888526, −2.88727216533985455992295871001, −2.27996323175163510710402796475, −1.00417591346342656048887968527, 0, 1.00417591346342656048887968527, 2.27996323175163510710402796475, 2.88727216533985455992295871001, 3.85325849305816087434107888526, 4.47651206452587388206130850645, 5.62171747222069653671744841533, 6.28798714251552491550955338860, 6.62894565150512462651818119117, 7.64934973439899225727472268030

Graph of the $Z$-function along the critical line